Nope. That's how it works in game as well. Intimidate happens immediately once a Pokemon hits the field and switch-in order is based on speed of fainted Pokemon.

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Actually, TV-Rocka is correct. In this case of a double faint, abilities will not activate until both Pokémon are on the field. So he should have gotten that Defiant boost from Intimidate (which would've allowed him to OHKO Landorus-T with Sucker Punch but Togekiss had Follow Me to stop Sucker Punch anyway).